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Preprint submitted on 8 Feb 2021
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On minimal kernels and Levi currents on weakly
complete complex manifolds
Fabrizio Bianchi, Samuele Mongodi
To cite this version:
Fabrizio Bianchi, Samuele Mongodi. On minimal kernels and Levi currents on weakly complete com-plex manifolds. 2021. �hal-03134521�
ON WEAKLY COMPLETE COMPLEX MANIFOLDS
FABRIZIO BIANCHI1 AND SAMUELE MONGODI2
Abstract. A complex manifold X is weakly complete if it admits a continuous plurisubharmonic exhaustion function φ. The minimal kernels ΣkX, k ∈ [0, ∞] (the
loci where are all Ckplurisubharmonic exhaustion functions fail to be strictly plurisub-harmonic), introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far X is from being Stein. We compare these notions, prove that all Levi currents are supported by all the ΣkX’s,
and give sufficient conditions for points in Σk
X to be in the support of some Levi
current.
When X is a surface and φ can be chosen analytic, building on previous work by the second author, Slodkowski, and Tomassini, we prove the existence of a Levi current precisely supported on Σ∞X, and give a classification of Levi currents on X.
In particular, unless X is a modification of a Stein space, every point in X is in the support of some Levi current.
1. Introduction
Given an abstract (and possibly very complicated) manifold, a natural question is whether it is possible to see it as a subset of a simpler space. In the real category, a fundamental theorem by Nash states that this is always possible, and in a very strong sense: every Riemannian manifold can be isometrically embedded in some RN. When moving to the complex category, we can then ask the following natural question: is it possible to embed any complex manifold in some CN, by means of a holomorphic map? We call Stein a manifold for which the above holds true. This time, the rigidity of holomorphic functions readily provides negative examples: for instance, the maximum principle implies that any holomorphic map on a compact complex manifold must be constant, and thus the manifold cannot be Stein. A central question is then to under-stand when a given complex manifold is Stein. More specifically, given a dimension n, one would like to understand the obstructions for an n-dimensional manifold to be Stein.
A major advance in this direction was provided by Grauert [3]: a complex manifold is Stein if and only if it admits a C2 strictly plurisubharmonic (psh for short) exhaustion function. The C2 assumption was relaxed to C0 by Narasimhan [12, 13]. In view of these results, it is natural to tackle the question by studying the positive cone Psh0e(X) of all continuous psh exhaustion functions on X (or more generally the cone Pshke(X) := Psh0e(X) ∩ Ck for some k ∈ [0, ∞], and in particular to find obstructions for them to be strictly psh. As a rough idea, such obstructions must correspond to the presence of some sets in X along which all continuous psh functions must necessarily be
Date: February 8, 2021.
2010 Mathematics Subject Classification. 32C40, 32E05, 32U10.
pluriharmonic. As a prototypical example, the blow-up of a point and its corresponding exceptional divisor give precisely this kind of obstruction.
A precise study of this kind of phenomena was started by Slodkowski and Tomassini in [24] in the setting of weakly complete complex manifolds, i.e., manifolds admitting a continuous psh exhaustion function. A crucial definition is the following: for k ∈ [0, ∞] the minimal kernel of a manifold X (with respect to Pshke) is
(1) ΣkX := {x ∈ X : i∂∂u is degenerate at x ∀u ∈ Pshke(X)},
i.e., the subset of X where no element of Pshke can be strictly psh. A key result of [24] is that, whenever Pshke(X) is not empty, there actually exists a function φ0 ∈ Pshke(X)
(called minimal ) which fails to be strictly psh precisely on the minimal kernel Σk X.
Moreover, the minimal kernels are local maximum sets (see Definition 3.1). Some finer properties are also established (some requiring at least the C2 regularity, see for instance [24, Theorem 3.9]). Observe that the ΣkX’s are increasing in k, but it is not known whether equalities should occur in general, see for instance [23, Section 5.10].
In [9, 10], the second author, Slodkowski, and Tomassini showed that, if X has complex dimension 2 and Psh∞e contains at least one real analytic function, the minimal kernel is either a union of countably many compact (and negative) curves or equal to the whole manifold, by giving a full classification of the possible structures that such a manifold can present. An important point here is that, although in general the minimal kernel does not have a priori an analytic structure, however its intersection with any level of a psh exhaustion function does (at least in dimension 2).
In [20], Sibony introduced the notion of Levi current (see Definition 2.1), which is related to the (non-)existence of strictly psh functions on a complex manifold and thus to the problem of determining whether a given manifold is Stein, see also [15, 19, 21]. Extremal Levi-currents are supported on sets where all continuous psh functions are constant. In the case of infinitesimally homogeneous manifolds, a foliation is con-structed and linked to the obstructions to Steinness.
Our goal here is to compare these two approaches, and in particular to use the notion of Levi current on X to study the analytic structure of the minimal kernels ΣkX. In order to do this, let us denote by Pshk, for 0 ≤ k ≤ ∞, the cone of Ck psh function on X and define the distribution Ek in T X as
(2) Ek := {(x, v) ∈ T X : (dϕ)x(v) = 0 ∀ϕ ∈ Pshk(X)}.
A distribution is a subset of T X whose intersection with TxX is a (real) vector subspace
of the latter for every x ∈ X. In general, Ek will not be a subbundle of T X, as dim Exk is not constant; however it is a closed subset of T X, hence the function x 7→ dim Exk is upper semicontinuous.
The following is our main result.
Theorem 1.1. Let X be a weakly complete complex manifold and T a Levi current on X. Denote by Σk
X the minimal kernels of X and by F the union of the supports of all
Levi currents on X. Then F ⊆ ΣkX for all k ≥ 0 and
(1) if T has compact support KT, then KT is a local maximum set;
(2) if K is a local maximum set, then there exists a Levi current supported on K. In particular, K ∩ F 6= ∅.
Moreover, if X is a surface, φ ∈ Ck a psh exhaustion function, and Y a regular connected component of a level set {φ = c},
(3) if 4 ≤ k ≤ ∞ and U ⊆ Σk
X is an open set in Y and there exists x ∈ U such
that dim Exk= 2, then X is a union of compact complex curves. In particular, F = ΣkX = X;
(4) if 2 ≤ k ≤ ∞ and Y ⊆ Σk
X, then there exists c0 < c such that the connected
component of φ−1([c0, c]) containing Y is contained in F . In particular, Y ⊆ F . Moreover, given any psh function u ∈ Psh0(X), any Levi current T can be naturally disintegrated as T = R Tcdµ, where µ is a positive measure on R and Tc is a Levi
current supported on {u = c}, see Corollary 2.3. This in particular gives examples of Levi currents for which Item (1) applies. Notice that, whenever two level sets of u, v ∈ Psh0(X) do not coincide, this allows to further refine the description of the extremal Levi current. This motivates the definition (2) of the distributions Ek.
It follows from [9, 24] that, when k ≥ 2, for any level set Y of an exhaustion function φ ∈ Psh0e, the set ΣkX ∩ Y is a local maximum set (or empty), see Lemma 3.4. Hence Item (2) applies for instance to such sets. Finally, the manifold X = C × P1 provides
an example where the Items (3) and (4) apply.
Remark 1.2. It would be interesting to know if the equality holds in Item (2) (and in particular for the intersections between levels sets of a psh exhaustion function and the minimal kernel). Namely if, for any point in a local maximum set K, there exists a Levi current T such that x ∈ spt T .
The paper is organized as follows. In Section 2 we recall the definition of Levi currents and the properties that we will need in the sequel. In Section 3 we prove Items (1) and (2) of Theorem 1.1. The first item is established for Σk
X ∩ {φ0 = c0}
(where c0 is an attained value for a continuous minimal function φ0 and k ≥ 2) in
[24, Theorem 3.6], and is actually a consequence of [19, Theorem 3.1], where it is proved through an integration by parts, see also [20, Proposition 4.2] for an analogous statement for F . We give here a different proof by means of a characterization of the local maximum property due to Slodkowski [22] which allows to bypass the use of Brebermann functions and Jensen measures as in [24]. In Section 4 we study the relation between the minimal kernels ΣkX and distributions in the tangent bundle T X given by directions satisfying some degeneracy condition. This leads to the proof of Item (3). The proof of Theorem 1.1 is completed in Section 5, where we establish Item (4). In Section 6 we consider the case where X is a surface and the exhaustion function in Theorem 1.1 can be chosen analytic. By exploting the main result in [9], we deduce a classification of Levi currents in this case.
Acknowledgements. The authors would like to thank Nessim Sibony for the refer-ences and for very useful comments that also helped to improve Item (2) in the main theorem, and Zbigniew Slodkowski for very helpful remarks on a preliminary version of this paper. They also would like to thank Viˆet-Anh Nguyˆen and Giuseppe Tomassini for useful observations and discussions.
This work was supported by the Research in Pairs 2019 program of the CIRM (Cen-tro Internazionale di Ricerca Matematica), Trento and the FBK (Fondazione Bruno Kessler). The authors would like to warmly thank CIRM-FBK for their support,
hospitality, and excellent work conditions. This project also received funding from the I-SITE ULNE (ANR-16-IDEX-0004 ULNE), the LabEx CEMPI (ANR-11-LABX-0007-01) and from the CNRS program PEPS JCJC 2019.
2. Levi currents on complex manifolds
In this section we recall the definition of Levi current and give the properties that we need in the sequel. These results are essentially contained in [20, Section 4] and [21, Section 3], we sketch here the proofs for completeness. We let X be any com-plex manifold and we denote by Psh0(X) the space of continuous plurisubharmonic functions on X.
Definition 2.1 (Sibony [20]). A current T on X is called Levi current if (1) T is non zero;
(2) T is of bidimension (1, 1); (3) T is positive;
(4) i∂∂T = 0;
(5) T ∧ i∂∂u = 0 for all u ∈ Psh0(X).
A Levi current T is extremal if T = T1 = T2 whenever T = (T1+ T2)/2 for T1, T2
Levi currents.
Lemma 2.2. Take u ∈ Psh0(X) and let T be a Levi current. The currents T ∧ ∂u , T ∧ ∂u , and T ∧ ∂u ∧ ∂u
are all well defined and vanish identically on X.
Proof. The currents in the statement are well defined when u is smooth, and the arguments from [2, Section 2] and [20, Section 4] prove the good definition for u ∈ Psh0(X).
If u ∈ Psh0(X), then also exp(u) ∈ Psh0(X). So, by Definition 2.1 of Levi current, T ∧ i∂∂ exp(u) vanishes identically. Hence, we have
0 = exp(u)T ∧ i∂∂u + i exp(u)T ∧ ∂u ∧ ∂u .
Given that T ∧ i∂∂u = 0, we conclude that T ∧ ∂u ∧ ∂u = 0. This gives the last identity. We prove now the first one, the proof for the second one is similar. Since T is positive, for any (0, 1)-form α by Cauchy-Schwarz’s inequality we get
|hT, ∂u ∧ αi|2≤ chT, ∂u ∧ ¯∂ui · hT, α ∧ ¯αi
where c is a constant independent of T, u, and α. Since the first factor in the RHS is
zero by the first part of the proof, the assertion follows.
By a standard disintegration procedure, we obtain the following consequence. Corollary 2.3. Suppose T is a Levi current and u ∈ Psh0(X); then there exists a measure µ on R and a collection of currents Tc, c ∈ R such that
• Tc is supported on Yc= {x ∈ X : u(x) = c} for all c ∈ R;
• Tc is non zero for µ-almost every c ∈ R;
• for every 2-form α on X we have hT, αi =
Z
R
hTc, αidµ(c) .
Moreover, if u ∈ Psh1(X) and c is a regular value for u, then Tc = j∗Sc, where j is
the inclusion of Yc in X and Sc a current on the real manifold Yc.
Notice that Tcneeds not be extremal, as it is easily seen considering X = C × P1.
Remark 2.4. Suppose now that X is weakly complete and let φ be a psh exhaustion function. By Corollary 2.3, every Levi current is obtained by averaging Levi currents which are supported on the level sets of φ; as the latter is an exhaustion of X, its level sets are compact, so every Levi current on X is an integral average of compactly supported Levi currents, i.e., positive currents of bidimension (1, 1) which are ∂∂-closed and compactly supported.
Corollary 2.5. If T is a Levi current and u ∈ Psh1(X), then the vector field associated to T is tangent to the kernel of ∂u∧∂u, whenever the latter is non-zero (and the former is defined). Moreover, if there exists v ∈ Psh0(X) which is strictly plurisubharmonic at a point x ∈ X, then x 6∈ spt T for any Levi current T .
Proof. The first statement is equivalent to T ∧ ∂u ∧ ∂u = 0, hence follows from Lemma 2.2. Suppose now that we have v ∈ Psh0(X) which is strictly psh at x and a Levi current T . First, by Richberg [16, Satz 4.3] we can assume that v is C∞ and strictly psh near x. Then, if ρ ∈ C0∞(X) is supported in a neighbourhood of x where v is strictly psh and kρkC2 is small enough, then also v + ρ is psh. In particular, we can choose
ρi, 1 ≤ i ≤ 2n such that the ker(∂(v + ρi) ∧ ∂(v + ρi))x are independent (over R). This
property holds true in a neighbourhood of x. Hence, as the vector field associated to T (on a full measure subset of the support T for the mass measure) should belong to all these subspaces, the only possibility is that x 6∈ spt T . This concludes the proof. Lemma 2.6. Let T be a Levi current such that K = spt T is compact. If u is defined and plurisubharmonic in an open neighbourhood V of K and strictly plurisubharmonic at x ∈ V , then x 6∈ spt T .
Proof. Let V0 b V be an open neighbourhood of K containing x. Let χ ∈ C0∞(V )
be such that χ|V0 ≡ 1, then χu is defined on X and psh on V0. As spt T ⊆ V0, also
spt(T ∧ i∂∂u) ⊆ V0, so T ∧ i∂∂u is positive; moreover, as T is a Levi current, we have i∂∂T = 0, hence
0 = hi∂∂T, ui = hT, i∂∂ui = hT ∧ i∂∂u, 1i. Therefore, T ∧ i∂∂u = 0 as a (positive) measure.
Since (i∂∂u)x > 0, this happens in a neighbourhood of x, so T ∧ i∂∂u is strictly
positive in a neighbourhood of x unless T is zero there. This gives x 6∈ spt T and
concludes the proof.
Lemma 2.7. Suppose that a current T satisfies requests 1 − 4 of Definition 2.1 and T has compact support. Then T is a Levi current.
Proof. Given that T is compactly supported, so are uT , T ∧ ∂u, T ∧ ∂u, and T ∧ i∂∂u for all u ∈ Psh0(X). Moreover, as T is positive and u is psh, T ∧ i∂∂u is a positive measure on X; therefore, it is zero if and only if hT ∧ i∂∂u, 1i = 0.
Notice that, by Stokes’ theorem, we have hi∂∂(uT ), 1 = 0i, hence
0 = −hi∂∂u ∧ T, 1i + hui∂∂T, 1i + hi∂(∂u ∧ T ), 1i − hi∂(∂u ∧ T ), 1i.
We have i∂∂T = 0 by hypothesis, while hi∂(∂u ∧ T ), 1i and hi∂(∂u ∧ T ), 1i vanish by another application of Stokes’ theorem. Therefore T ∧ i∂∂u = 0, that is, T is a Levi
current.
3. Local maximum sets
We establish here Items (1) and (2) of Theorem 1.1. We recall the following defini-tion, see also [24, Section 2] and [17].
Definition 3.1. Let X be a complex manifold and K ⊂ X be compact. We say that K is a local maximum set if every x ∈ K has a neighbourhood U with the following property: for every compact set K0 ⊂ U and every function ψ which is strictly psh in a neighbourhood of K0, we have
max
K∩K0ψ = maxK∩bK0ψ.
Proposition 3.2. Suppose that X is weakly complete. If T is a Levi current with compact support, then spt T is a local maximum set.
Proof. Suppose that K := spt T is not a local maximum set. By [22, Proposition 2.3] there exist x ∈ K, a neighbourhood B of x, with local coordinates z with origin in x, such that B ≡ {z ∈ Cn : kzk < 1}, and ψ : B → R strictly psh with ψ(x) = 0 and ψ(y) ≤ −kz(y)k2 for all y ∈ K ∩ B. Up to replacing ψ by an element of a continuous
approximating sequence, we can directly assume that ψ is continuous. By taking a possibly smaller ball a x, we can also assume that −kz(y)k2− /8 ≤ ψ(y) on K. Set
A := {y ∈ B : kz(y)k2> 3/4} and V = {y ∈ B : |ψ(y) + kz(y)k2| < /4}. By the continuity of ψ and the bounds above, V is an open subset of B containing K. Consider u ∈ Psh0(X) such that u(x) = −/4 and supB|u| < /2 (this function exists because of the assumption on X). Since K ∩ B ⊂ V , there also exists χ ∈ C0∞(X) be such that χ|K≡ 1 and spt χ ∩ B ⊂ V . Define the function v : X → R as
v = (
χ max{u, ψ} on B,
χu on X \ B.
We claim that v = χu on A. Indeed, for every p ∈ spt χ ∩ A, we have that p ∈ V , and so ψ(p) < −3/4 + /4 = −/2. Hence, ψ(p) < u(p) and v(p) = (χu)(p).
By construction, v coincides with ψ in a neighbourhood of x. It follows that v is psh in a neighbourhood of K and strictly psh in x. Therefore, we have x 6∈ spt T by Lemma 2.6. This gives a contradiction with the choice of x ∈ K and completes the
proof.
Proposition 3.3. Let K ⊂ X be a local maximum set. There exists a Levi current T such that spt T ⊆ K.
Proof. By [21, Theorem 3.1] (see also [20, Section 4]) and Lemma 2.7, if there are no Levi currents supported on K, there exists a smooth strictly psh function u on some open neighbourhood U of K. By slightly perturbing u, for every x0 ∈ K we
U such that du1, . . . , du2n are linearly independent at x0. This implies that, in a
neighbourhood of x0, we have
{u1 = u1(x0)} ∩ . . . {u2n= u2n(x0)} = {x0}.
By [23, Corollary 1.11] and [22, Theorem 4.2], for every family of continuous psh functions on U0 there exists a local maximum set K0 ⊆ K with the property that all functions of the family are constant on K0. Choosing a point x0 ∈ K0, the previous
paragraph gives that x0 is isolated in K0. This is a contradiction, and the proof is
complete.
We conclude the section with the following result, that we will need to prove Item (3) of Theorem 1.1.
Lemma 3.4. Let X be a weakly complete complex surface and Y a regular level for a C0 exhaustion function φ. Then, for all k ≥ 2, Σk
X∩ Y is a local maximum set and, for
all local maximum sets K ⊆ ΣkX, K is foliated by holomorphic discs, i.e., it is locally a union of disjoint holomorphic discs.
Proof. The intersection ΣkX ∩ Y is a local maximum set by [9, Theorem 3.2]. As observed in [9, Proposition 3.5], the proof of the lemma is then essentially given in [24, Lemma 4.1], see in particular the Assertion in the proof of that lemma. We point out that [24, Lemma 4.1] relies on a result by Shcherbina [18], which holds true only in dimension 2; this is the reason for restricting ourselves to the case of surfaces. it would be interesting to prove (or disprove) a similar statement in higher dimension.
4. Kernels and tangent directions
In this section we let X be a weakly complete complex manifold of dimension n and assume that Pshk(X) contains at least one exhaustion function φ : X → R for some 2 ≤ k ≤ ∞. Recall that the minimal kernel ΣkX of X is defined as in (1) and the distribution Ek of T X as in (2). We consider further the distribution Sk of T X given by
(3) Sk:= {(x, ξ) ∈ T X : ξ ∈ TxX, (i∂∂u)x(ξ, ξ) = 0 ∀u ∈ Pshk(X)} .
Similar objects have already appeared in relation to the study of the Levi problem, see for instance [5] in the case of homogeneous manifolds and [4, 24]. We also set
Ek` := {x ∈ X : dim Exk≥ `} and S1k:= {x ∈ X : dim Sxk≥ 1}. By definition, E`k ⊆ Ek
`−1 and E`k is closed in E`−1k for all ` ≥ 1. Observe moreover
that Sk is a complex distribution.
Remark 4.1. Let T be a Levi current. Then, for almost every point of the support of T (with respect to the mass measure), the vector field associated to T at x belongs to the fibre Sxk of Sk at x.
Proposition 4.2. We have Sk ⊆ Ek, and Sk
Proof. It follows from the definition of Ek that E1k = E2k. Moreover, if (x, v) 6∈ Ek, there exists ψ ∈ Pshk(X) such that (dψ)x(v) 6= 0; then
i∂∂ exp(ψ) = exp(ψ)i∂∂ψ + i exp(ψ)∂ψ ∧ ∂ψ ≥ 0,
which implies that (i∂∂ exp(ψ))x(v, v) ≥ exp(ψ(x))|∂ψx(v)|2 > 0 and so (x, v) 6∈ Sk.
It follows that Sk ⊆ Ek.
We now prove that Ek1 = ΣkX. If x 6∈ ΣkX, then there exists ψ ∈ Pshke(X) which is strictly psh around x; therefore, given any ρ : X → R smooth with compact support near x, there exists > 0 such that ψ+ρ is still psh. So, we can construct psh functions of class Ck whose differentials span the tangent space at x, which implies that these differentials do not have any nontrivial common kernel in TxX. So Exk = {0}, hence
Ek
1 ⊆ ΣkX.
On the other hand, if Exk= {0}, given v1, . . . , v2n∈ TxX linearly independent, we can
choose psh functions ψij, i, j = 1, . . . , 2n of class Ckand such that (dψij)x(vi+ vj) 6= 0.
Therefore, the function ψ := P2n
i,j=1ψ2ij has positive defined Levi form at x. Adding
to the exhaustion function φ suitable multiples of ψ, we see that x 6∈ ΣkX. This gives E1k ⊇ Σk
X, hence E1k = ΣkX.
In order to conclude, we need to prove that Sk
1 ⊇ ΣkX. Take x ∈ ΣkX and suppose
by contradiction that, for every v ∈ TxX there is ϕv : X → R which is Ck, psh, and
such that (ddcϕ
v)x(v, v) > 0. Then, as above, we can construct a Ck function ψ which
is strictly psh at x. This gives the desired contradiction and completes the proof. The following result gives Item (3) of our main Theorem 1.1.
Proposition 4.3. Let X be a weakly complete complex surface, φ a Ck, 4 ≤ k ≤ ∞,
exhaustion psh function and Y a regular connected component of a level set {φ = c} of φ. Suppose that U ⊆ Y is an open set in Y and U ⊆ ΣkX. If there exists x ∈ U such that dim Exk = 2, then X is a union of compact complex curves. In particular, ΣkX = F = X = E2k, and E3k is contained in a (possibly empty) analytic subset of the singular levels for φ.
We will need the following theorem by Nishino, see [14, Proposition 9 and Th´eor`eme II] and [11, Section 2.2.1].
Theorem 4.4 (Nishino). Let X be a weakly complete or compact surface that contains an uncountable family F of disjoint connected compact complex curves. Then there exist a Riemann surface R and a meromorphic map h : X → R with compact fibers. Proof of Proposition 4.3. By Theorem 4.4, to prove the first assertion it is enough to show that X contains uncountably many disjoint compact complex curves.
Since x ∈ U is such that dim Exk = 2, there exists ψ ∈ Pshke(X) such that (dφ)x
and (dψ)x are linearly independent; hence, the map ψ|Y : Y → R is not constant.
Since k ≥ 4, by Sard’s theorem we can find regular values b for ψ arbitrarily close to b0 := ψ(x), therefore the sets Cb = {y ∈ Y : ψ(y) = b} intersect the open set
U ⊆ ΣkX.
For any y ∈ Cb∩ΣkX, by Proposition 4.2 we have TyCb = Eyk= Syk. Therefore, Cb∩ΣkX
is a complex curve, being a real, smooth 2-dimensional manifold with complex tangent space. On the other hand, the set Cb\ ΣkX is open in Cb. Let z ∈ Cb be a boundary
f : D → X such that f (D) ⊂ Y and f (0) = z. If ζ ∈ D is close enough to 0, then, setting w = f (ζ), we have w ∈ ΣkX, and (dφ)w and (dψ)w independent. This gives
w ∈ E2k\ Ek
3, which in turn implies that Ewk = TwCψ(w). Therefore f (D) coincides
locally with a leaf Cb0. Hence Cb is contained in Σk
X, so it is a compact complex curve.
As b was taken arbitrarily among the regular values close enough to b0, we find
uncountably many disjoint (since they correspond to distinct values) compact complex curves in X, as desired.
In order to conclude, we need to prove the final assertion on E3k. We proved above that there exists a meromorphic map h : X → R with compact fibres, where R is Riemann surface. It is enough to prove that E3k ⊆ {h0 = 0}.
Let x ∈ X be such that h0(x) 6= 0. Consider a strictly psh exhaustion function ψ for R (which we can assume to be C∞ near x by [16]) and the family of functions F := {ψ + ρ}, where ρ is a smooth function compactly supported near h(x). For every such ρ, ψ + ρ is still strictly psh for sufficiently small. Thus, we can obtain a set of generators for the tangent space given by differentials at h(x) of psh functions in F . Pre-composing the corresponding functions with h, we obtain that the space of differentials at x of psh functions on X has dimension at least 2. Hence, x /∈ Ek
3, and
the proof is complete.
Remark 4.5. Suppose that X is a surface and Y a regular level for an exhaustion function φ ∈ Psh0e(X). Let K ⊆ Y ∩ ΣX be a local maximum set. By Lemma 3.4,
Y and K are foliated by holomorphic discs. For every such disk, its tangent bundle is exactly the restriction of S. By [1, Theorem 1.4], there exists a ∂∂-closed positive current of bidimension (1, 1), directed by S, supported in K. By Lemma 2.7, such current is a Levi current. This gives a different proof of Item (2) when dim X = 2.
5. End of the proof of Theorem 1.1
It follows from Corollary 2.5 (or Lemma 2.6) that spt T ⊆ ΣkX for every Levi current T and all k ≥ 0. Thus, we have F ⊆ ΣkX for all k ≥ 0. Moreover, Items (1), (2), and (3) follow from Propositions 3.2, 3.3, and 4.3, respectively.
Let now Y be a regular connected component of a level set for an exhaustion psh function φ ∈ Pshke(X) for some k ≥ 2. The remaining item follows from the next proposition.
Proposition 5.1. If k ≥ 2 and Y ⊆ ΣkX, there exists c0 < c such that the connected component of φ−1([c0, c]) containing Y is contained in F .
Proof. We assume for simplicity that the level {φ = c} is regular and connected, the argument is similar otherwise. Since k ≥ 2, by [24, Theorem 3.9] there is c0 < c such that, setting
K = {x ∈ X : c0 ≤ φ(x) ≤ c},
the form (ddcφ)2 vanishes on the interior of K, hence on K. So, we have K ⊆ ΣkX. Consider the current T given by
T := i∂φ ∧ ∂φ.
It is clear that T is a current of bidimension (1, 1), positive and directed by the complex subspace of the tangent of the levels of φ. Moreover, i∂∂T is induced by the form
So, i∂∂T vanishes where φ is not strictly psh, hence on ΣkX. Let B be the interior of K, then the restriction of T to B is a current of bidimension (1, 1), positive, ∂∂-closed (in B); moreover, given u ∈ Psh0(X), we have that T ∧ ∂∂u = 0 on ΣkX, so T is a Levi current.
By construction and Lemma 2.2 we have T ∧ ∂φ = 0, so we can disintegrate T along the levels of φ, see Corollary 2.3: there exist currents Ts with s ∈ (c0, c), such that, for
α a 2-form with spt α ⊂ B, hT, αi =
Z c
c0
hTs, αi dµ(s) for some measure µ on (c0, c).
Since φ ∈ C2, the measure µ is absolutely continuous with respect to the Lebesgue
measure on (c0, c).
As T is ∂∂-closed in B, so is µ-almost every Ts in B; therefore, for a dense open set
of s ∈ (c0, c), Ts is a positive, ∂∂-closed current of bidimension (1, 1) and
spt Ts= {x ∈ X : φ(x) = s}.
The set in the RHS is compact since φ is an exhaustion function. By Lemma 2.7, Ts
is a Levi current.
In conclusion, the level set {x ∈ X : φ(x) = s} is contained in F for almost all s ∈ (c0, c), so φ−1([c0, c]) ⊆ F , as F is closed. In particular, Y ⊆ F .
The proof of Theorem 1.1 is complete.
6. Real analytic exhaustion function
A classification of those weakly complete complex surfaces X admitting an analytic exhaustion function is given in [9]. As a direct consequence, we can get an analogous complete classification of the possible Levi currents in this setting.
First notice that each exceptional divisor V in X corresponds to an extremal Levi current given by the current of integration [V ]. Without loss of generality, to simplify our next statement, we can thus assume that X has no such divisors on the regular levels of α. The statement for a general X is then a direct consequence.
Theorem 6.1. Let X be a weakly complete complex surface admitting an analytic exhaustion function α. Assume that X has no exceptional divisors on the regular levels of α. Then one of the following possibilities hold:
(1) X is Stein (and so, admits no Levi currents);
(2) F = Σ∞X = X = ∪Vi, where all the Vi are (disjoint) connected compact curves,
and all extremal Levi currents are of the form λ[Vi0] for some positive λ, with Vi0 an irreducible component of some Vi;
(3) F = Σ∞X = X, every regular level Yc of α is foliated by curves Ui, and the
support of any extremal Levi currents on Ycis equal to (a connected component
of ) Yc.
Observe also that, although a priori we would only have ΣkX ⊆ Σ∞X for all k ≥ 0, the above geometric description implies that ΣkX = Σ∞X for all k ≥ 0.
Proof. It follows from [9, Theorem 1.1] that one of the following possibilities holds: (1) X is a Stein space;
(2) X is proper over a (possibly singular) complex curve;
(3) the connected components of the regular levels of α are foliated with dense complex curves.
In the first and second cases, the assertion follows from the characterization of Levi currents given in Section 2. In the third case, a Levi current can be constructed, for instance, by means of [1, Theorem 1.4]. By proposition 3.2, the support of any Levi current is a local maximum set. By [7, Lemma 3.3], a local maximum set contained in a Levi-flat hypersurface must be a union of leaves of the Levi foliation.
Hence, in the third case, any Levi current on a regular level set of the exhaustion function is supported on the whole level set, as all the leaves of the Levi foliation are dense. This in particular applies to extremal Levi currents. The proof is complete.
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1
CNRS, Univ. Lille, UMR 8524 - Laboratoire Paul Painlev´e, F-59000 Lille, France Email address: [email protected]
2
Politecnico di Milano, Dipartimento di Matematica, Via Bonardi, 9 – I-20133 Milano, Italy
